Monomial ideals with arbitrarily high tiny powers in any number of variables

Abstract

Powers of (monomial) ideals is a subject that still calls attraction in various ways. Let $I\subset \mathbb K[x_1,\ldots,x_n]$ be a monomial ideal and let $G(I)$ denote the (unique) minimal monomial generating set of $I$. How small can $|G(I^i)|$ be in terms of $|G(I)|$? We expect that the inequality $|G(I^2)|>|G(I)|$ should hold and that $|G(I^i)|$, $i\ge 2$, grows further whenever $|G(I)|\ge 2$. In this paper we will disprove this expectation and show that for any $n$ and $d$ there is an $\mathfrak m$-primary monomial ideal $I\subset \mathbb K[x_1,\ldots,x_n]$ such that $|G(I)|>|G(I^i)|$ for all $i\le d$.